The Frobenius problem for generalized repunit numerical semigroups

TL;DR

This paper introduces a new family of numerical semigroups generated by generalized repunit numbers, providing explicit formulas for the Frobenius number and other invariants, and establishing their properties like homogeneity and Wilf's conjecture.

Contribution

It defines a broad class of semigroups generated by generalized repunits and derives explicit formulas for their Frobenius number and invariants, extending previous work on repunit semigroups.

Findings

Derived a closed-form formula for the Frobenius number.

Proved these semigroups are homogeneous and satisfy Wilf's conjecture.

Computed invariants such as Apéry sets, genus, and type.

Abstract

In this paper, we introduce and study the numerical semigroups generated by $\{a_1, a_2, \ldots \} \subset \mathbb{N}$ such that $a_1$ is the repunit number in base $b > 1$ of length $n > 1$ and $a_i - a_{i-1} = a\, b^{i-2},$ for every $i \geq 2$, where $a$ is a positive integer relatively prime with $a_1$. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of $a, b$ and $n$, and compute other usual invariants such as the Ap\'ery sets, the genus or the type.